Abstract: Let B(l_p,l_q) denote the bounded linear operators from l_p to l_q, for 1 ≤ p, q ≤ ∞. Termwise multiplication of two bounded linear operators in B(l_p,l_q) yields another bounded linear operator in B(l_p,l_q), called their Schur product (and sometimes misnamed their Hadamard product). The Schur product thus defines a commutative Banach algebra structure on B(l_p,l_q), even though when p ≠ q there was no natural product on this space, and when p = q there is a natural product but it is not commutative.

The maximal ideal spaces of these algebras are shown to be points in the Stone-Čech compactification of N x N, where N denotes the natural numbers. Various ideals and their hulls are examined, and several open questions are posed.

Keywords: Schur multiplication, Banach algebra, maximal ideal space, Stone-Cech compactification, operator theory, l_p spaces, spectral synthesis

Complete paper. This paper appears in Journal of Operator Theory 5 (1981), pp. 231-243.

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